A high-throughput wireless-powered relay network with joint time and power allocations

Computer Networks 160 (2019) 65–76

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A high-throughput wireless-powered relay network with joint time and power allocations Gaofei Huang a,∗, Wanqing Tu b a b

Faculty of Mechanistic and Electric Engineering, Guangzhou University, Guangzhou, China School of Computer Science, The University of Auckland, Auckland 1142, New Zealand

a r t i c l e

i n f o

Article history: Received 24 December 2018 Revised 24 April 2019 Accepted 5 June 2019 Available online 8 June 2019 Keywords: Radio-frequency (RF) energy harvesting Simultaneous wireless information and power transfer (SWIPT) Relay Dynamic programming

a b s t r a c t This paper studies how to maximise the throughput for a wireless-powered relay network where an unreliable direct link exists between a source node and a destination node. Our idea is to optimally combine time allocations (for different relay activities) and power allocations (for the relay to forward different frames) based on a harvest-store-consume (HSC) model. With the HSC model, the network may operate in a direct transmission (DT) mode or a cooperative transmission (CT) mode to transmit one frame. Accordingly, the relay may either harvest energy from the radio-frequency signals in the direct link in DT mode, or harvest energy and forward information in CT mode, where the energy harvested in both modes is stored temporarily before it is consumed for information relaying. Moreover, in our CT mode, the relay distributes the energy to the transmissions of multiple consecutive frames by taking time-varying wireless channel conditions in a time period into account in order to achieve the best throughput performance. We formulate a deterministic optimization problem and a stochastic programming problem under the assumption of full channel state information (CSI) and causal CSI, respectively. Both problems optimally determine DT or CT modes for each frame transmission. In the CT mode, the problems also decide the time-switching (TS) ratios and the transmit power at the relay. Our formulated problems are intractable due to the energy distribution at the relay between multiple frames. This intractability is addressed by dynamic programming techniques which decompose the problems into two tractable subproblems: an outer problem, and an inner problem. By solving the two subproblems, we propose the joint TS operation and power allocation algorithms for the two original formulated problems. The proposed algorithm for causal CSI is an online algorithm with low complexity, while the proposed algorithm for full CSI is offline and provides a benchmark for the online algorithm. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Many wireless devices are energy-constrained wireless nodes. In order to extend the lifetime of these communication devices, studies have investigated simultaneous wireless information and power transfer (SWIPT) schemes in various wireless communication scenarios [1–9], allowing wireless devices to harvest energy from radio-frequency (RF) signals when they receive information. More specifically, in a wireless relay network with SWIPT, the relay node with RF energy-harvesting (EH) capability can not only receive information but also harvest energy via the RF signals emitted by the source. By using the harvested energy, the relay can

∗

Corresponding author. E-mail addresses: [email protected] (G. Huang), [email protected] (W. Tu). https://doi.org/10.1016/j.comnet.2019.06.005 1389-1286/© 2019 Elsevier B.V. All rights reserved.

be powered to forward information the destination, extending the lifetime of its batteries. 1.1. Related works Wireless-powered relaying has been studied for different wireless relay networks in recent years [10–24]. However, most studies have focused on a topology where a direct link between the source and the destination does not exist [10–22]. In practice, a direct link usually exists in a wireless relay network, although such a link may not perform well due to obstacles etc. With the existence of the direct link, when the source sends information to the destination directly, the relay may harvest energy from the RF signals in the direct link. However, it is not a trivial task to make a decision for the relay when to participate in the cooperative transmissions between the source and the destination. On one hand, if a small portion of time within a period is allocated to direct transmission, the relay harvests a small amount of energy and the performance

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of WPRN achieved by information relaying may be degraded instead of being improved since the relay forwards information with repetition coding strategy in two orthogonal phases of one frame transmission. On the other hand, if a large portion of time in a period is allocated to direct transmission, few information transmissions take place via the relay link. This results in that the achieved throughput is similar to that only achieved by direct transmission. Therefore, by taking the direct link into account, it is more complicated and challenging to implement design and analysis for the wireless-powered relay network (WPRN). Related studies have been carried out on such WPRNs recently. In [23], the authors study the WPRN with a direct link by using a harvest-then-forward (HTF) model, which is an energy consumption model based on the popularly studied conventional timeswitching relaying (TSR) or power-splitting relaying (PSR) protocol [10]. In the HTF model, the relay firstly employs a time-switching (TS) or power-splitting (PS) EH receiver to harvest energy, and then uses up all available energy for each frame transmission. In order to improve the throughput performance, the work [23] optimizes the TS ratios and the PS ratios for TSR-based and PSR-based WPRNs with rateless coding technology. However, the HTF model has two major drawbacks. Firstly, the harvested energy, only from a portion of the RF energy emitted by sending a frame, is too limited to gain significant performance improvement. Secondly, the relay is restricted to use up all harvested energy for each frame transmission. In fact, if the relay-to-destination (RD) link quality is poor, the energy can be utilized inefficiently as the transmission quality via such a link is not acceptable anyway. To overcome the drawback that the amount of harvested energy is quite small in the HTF model, the work [24] studies a WPRN with a direct link based on an accumulate-then-forward (ATF) model, where the relay assists in forwarding information only when the harvested energy reaches a predefined energy threshold and the destination fails to decode the information received from the source. However, the scheme makes the relay to transmit without considering the channel condition of the RD link, which is similar to that in the HTF model and may cause that the harvested energy is utilized inefficiently. 1.2. Motivations To address the drawbacks of the ATF and HTF models, we investigate how to optimally utilize both direct transmission and wireless-powered relaying for the WPRN in this paper. Our goal is to achieve the best average throughput performance in a time period for transmitting multiple frames. To achieve our goal, our idea is to optimally allocate time between direct transmission and wireless-powered relaying within a period of multiple frames by employing a new energy consumption model, namely harveststore-consume (HSC) model, which is different from the ATF and HTF models. The HSC model was first studied in [25] in a pointto-point wireless communication system where a source node harvests energy from environment (e.g., solar energy) and then reserves the energy for better use in the future. With the HSC model, the relay in our WPRN can store the harvested energy temporarily when the source sends information to the destination directly, i.e., the WPRN can operate in a direct transmission (DT) mode. Moreover, when the source sends information to the destination via the relay, the relay can adjust the amount of energy consumed for the transmission of one frame according to the channel conditions of the WPRN in a time period which includes consecutive multiple frame transmissions, instead of using up all available energy for each transmission. Therefore, compared with that in the ATF or HTF model, the relay in the HSC model can harvest and consume energy more flexibly, and thus the performance of the WPRN may be potentially improved.

In this paper, we assume that the relay is equipped with a TS EH receiver, since the current commercial circuits are designed to deliver information and harvest energy separately [26]. Accordingly, the WPRN can operate in DT mode or cooperative transsmission (CT) mode by enabling the relay to harvest energy or cooperate in information relaying in a TS fashion. We investigate how to jointly optimize the TS operation and transmit power at the relay, which enables the relay to efficiently collect energy and then optimally distribute the stored energy to the transmissions of different frames, eventually helping to maximize the throughput of WPRNs. 1.3. Contributions In the design of joint time and power allocations for the WPRN with a direct link, our contributions are concluded below. • We derive the upper bound of the throughput performance for a WPRN with a direct link. This bound is achieved in an offline manner by computer evaluations, where full channel state information (CSI) are generated according to the statistical information of the wireless channels before information transmissions. Because non-causal CSI is required, it is impractical. Nevertheless, it provides a benchmark for the performance which can be achieved in the case of causal CSI. The formulated optimization problem is to optimally determine DT or CT mode for the WPRN and the TS ratios at the EH receiver and transmit power at the relay in CT mode. It is deterministic but challenging to be solved as the involved energy scheduling at the relay is coupled over multiple frame transmissions. We employ dynamic programming and problem decomposition techniques to decouple it in time to obtain two subproblems: an outer problem, and an inner problem, which can be solved by a brute force approach and a two-level optimization approach, respectively. • We design an online algorithm for the practical case in which only causal CSI is available. The formulated problem, more difficult than the deterministic problem with full CSI, is a stochastic program. Based on the derivation of the offline algorithm, by using a finite-state Markov channel (FSMC) model [31], we derive a low-complexity online algorithm. This algorithm allows our network to obtain a throughput close to the upper bound of the offline algorithm. • We then use computer simulations to verify our theoretical studies. Our evaluation suggests that the throughput achieved by our algorithm is much higher than that based on the ATF or HTF model. Moreover, our evaluation also observes that the throughput can be improved significantly by efficiently ultilizing the direct link in the WPRN. The rest of the paper is organized as follows. Section 2 describes our WPRN with the HSC model. Section 3 designs the algorithm that achieves joint time switching and power allocation based on full CSI. Section 4 is about the algorithm achieving joint time switching and power allocation based on causal CSI. Simulation results are shown in Section 5. Section 6 concludes the paper. 2. System model We consider a three-node two-hop WPRN as shown in Fig. 1. The source node (N1) sends information to the destination node (N3) via the EH relay (N2) that is equipped with a TS receiver. The three nodes are equipped with a single antenna and operate in a half-duplex manner. We assume that the direct link between the source and the destination exists but it is unreliable due to obstacles, which is more practical than the scenario which assumes such a direct link does not exit. The source is powered by fixed supply, but the EH relay is powered by batteries. To prolong the life of the

G. Huang and W. Tu / Computer Networks 160 (2019) 65–76

Fig. 1. A two-hop wireless-powered relay network with unreliable direct link.

batteries, the relay only consumes the energy which is harvested from the RF signals emitted by the source while forwarding information to the destination. Moreover, to enable the relay to operate with the HSC model, we assume that the relay is equipped with a rechargeable battery or a supercapacitor to store the energy temporarily, which will be consumed later. Note that circuit power consumed by information decoding at the relay cannot be ignored in practice, since the circuit power reduces the net harvested energy that can be stored in the battery for future use [26,27]. Nevertheless, to simplify the expression of transmission strategy derivation and shed light on how the direct link affects the network performance, we assume that the circuit power can be ignored, which is also adopted in [10–24]. We will demonstrate that the development of our transmission algorithm for the WPRN can be easily extended to the scenario where the circuit power is taken into account (see Remark 1 at the end of Section 2.4). Meanwhile, with the current studies on multi-antenna devices, based on [6–8], we believe that the performance of wireless powered networks can be further improved. We will verify this insight in our future study. 2.1. HSC-based cooperative transmission With the HSC model, the whole process of communication in the considered WPRN takes place on a time period basis. Each time period includes the transmissions of K frames, whose indexes belong to a set denoted as K  {1, · · · , K }, and all frame transmissions are of the same duration Tf . The structure of each frame can be one of the two types as illustrated in Fig. 2. The first type is corresponding to the DT mode for the WPRN, where the source directly transmits information to the destination, while the relay harvests energy from the RF signals over the direct link and stores the energy in the rechargeable device. The second type is corresponding to the CT mode, where one frame is divided into three phases. In the first phase, the source, the relay and the destination perform the same operations as those in DT mode, re-

Fig. 2. Structures of frames transmitted by the DT mode and the CT mode.

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spectively. In the second phase, the source broadcasts information, while both the relay and the destination receive and decode the information. In the third phase, both the source and the relay cooperatively transmit information to the destination. We denote the ratios of the three phases in frame k as λ1 (k), λ2 (k) and λ3 (k), respectively, k ∈ K. Let λ(k ) = {λ1 (k ), λ2 (k ), λ3 (k ) } and define  where  (k )  λ(k )| 3i=1 λi (k ) = 1, λi (k ) ∈ [0, 1], i ∈ {1, 2, 3} . Note that with HSC model, the relay may consume a portion of the harvested energy to transmit, which means that some energy can be stored in the rechargeable device for future usage after each transmission at the relay. To achieve superior throughput performance in the WPRN, we assume that rateless coding technique is adopted when the WPRN operates in CT mode. That is, the source and the relay need to encode the information with rateless codes before they transmit to the destination. Moreover, similar to the related works on rateless coding based wireless relay networks [23,28,29], we assume that CDMA is employed. Then, in the second phase, the source transmit the encoded information to the relay and destination with its assigned spreading code, while the relay and the destination respectively try decoding the information by accumulating mutual information from the information stream received from the source. When the entropy of original information is marginally surpassed by the collected coded bits, the original information can be successfully decoded and recovered at the relay and the destination [23,28,29]. As the WPRN operates in CT mode, the quality of the source-to-relay link must be better than that of the direct link such that the relay can successfully decode the information before the destination, which will be proved later (see Proposition 1). After the relay decodes the information, it re-encodes the information with a rateless code and transmits it to the destination in the third phase. Meanwhile, the source continuously transmits information to the destination, since CDMA is employed and the orthogonal transmissions between the source and the relay in the third phase can be realized. Based on the received signals in the second and third phases, the destination may decode the information with the energyaccumulation (EA) method or the information-accumulation (IA) method [23,28,29]. For the EA-based receiving method, the source and the relay use the same spreading code and the same rateless code, while the destination employs the Rake receiver to receive information based on maximal ratio combining method. For the IA-based receiving method, the source and the relay use different spreading codes and independent rateless codes, while the destination can separate the information from the source and the relay to accumulate information. In this paper, we assume that the EA method is adopted, and the results derived in this paper can be easily extended to the WPRN where the IA method is adopted. We let θ (k) be indicator of WPRN operation mode. That is, we set θ (k ) = 0 for the WPRN in DT mode and θ (k ) = 1 for the WPRN in CT mode. Accordingly, the relay can determine its TS operation according to the values of θ (k) and λ(k), ∀k ∈ K. Specially, consider two adjacent frames, e.g., frame k and frame (k + 1 ) for k ∈ Ko  {1, · · · , K − 1}. Then, the TS operation at the EH receiver of the relay can happen as the following four cases:

1) θ (k ) = θ (k + 1 ) = 0, which means that the WPRN operates in DT mode in both of the two adjacent frames. In this case, the EH receiver only harvests energy, and thus TS operation does not happen. We illustrate this case in Fig. 3(a). 2) θ (k ) = 0 and θ (k + 1 ) = 1, which means that the WPRN operates in DT mode and CT mode in frame k and frame (k + 1 ), respectively. In this case, the EH receiver switches its operation at the end of the first and second phases in frame (k + 1 ) based on the value of λ(k + 1 ). We illustrate this case in Fig. 3(b).

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Fig. 4. Structure of the HSC-based relay.

ration of one frame, but are i.i.d. for different frames. For frame k, denote the channel coefficient of the link between the source and the relay as a1 (k), denote the channel coefficient of the link between the relay and the destination as a2 (k), and denote the channel coefficient of the link between the source and the destination as a3 (k). The corresponding channel gain is expressed as gi (k ) = |ai (k )|2 , where i ∈ {1, 2, 3}. Denote the received additive white Guassian noise (AWGN) power at the relay and the destination as σ22 (k ) and σ32 (k ), respectively. Then, the normalized chan-

g1 ( k ) , and the normalized σ22 (k ) g (k ) channel gains of g2 (k) and g3 (k) are expressed as G2 (k ) = 22 and σ3 ( k ) g3 ( k ) G3 (k ) = 2 , respectively. σ3 ( k )

nel gain of g1 (k) is expressed as G1 (k ) = Fig. 3. TS operation in two adjacent frames at the EH receiver of the relay for different θ (k) and θ (k + 1 ) (IR: information reception; IT: information transmission).

3) θ (k ) = 1 and θ (k + 1 ) = 0, which means that the WPRN operates in CT mode and DT mode in frame k and frame (k + 1 ), respectively. In this case, the EH receiver switches its operation at the end of the three phases in frame k based on the value of λ(k). We illustrate this case in Fig. 3(c). 4) θ (k ) = θ (k + 1 ) = 1, which means that the WPRN operates in CT mode in both of the two adjacent frames. In this case, the EH receiver switches its operation at the end of each phase in both of the two adjacent frames based on the values of λ(k) and λ(k + 1 ). We illustrate this case in Fig. 3(d). To enable the HSC-based cooperative transmission, the structure of the relay node is illustrated in Fig. 4. According to Fig. 4, there are two switches, i.e., S1 and S2 , to implement the TS operations between EH and information process at the relay, which is determined by the output of the TS decision unit, i.e., O1 and O2 . When the relay harvests energy, S1 switches to the EH branch and S2 is opened, which results in that the havested energy is stored in the battery. When the relay receives and forwards information, S1 switches to the transceiver branch and S2 is closed. Meanwhile, the TS operations between information receiving and information forwarding are determined by the output O3 of the TS decision unit, i.e., the values of λ2 (k) and λ2 (k) when the relay operates in CT mode in frame k.

In this paper, we consider two scenarios for the knowledge of CSI at the transmitters. Firstly, we consider an ideal scenario, where the transmitters have the knowledge of full CSI of K frames at the beginning of the first frame transmission in a period, which indicates that non-causal CSI is available. Then, we consider a practical scenario, where the transmitters have only the knowledge of current CSI before transmission, which indicates that only causal CSI is available. Moreover, for the second scenario, we assume that the statistics, i.e., probability density functions (pdfs), of the channels in the WPRN have been known. 2.3. Definition of throughput for the WPRN

In the considered WPRN, we assume that the source transmits with constant power, i.e., P1 , for all frames. Then, we can obtain the achievable end-to-end data rate for the WPRN in DT mode as RDT (k ) = R3 (k ) = log (1 + P1 G3 (k ) ). Moreover, we can obtain the achievable data rate from the source to the relay as R1 (k ) = log (1 + P1 G1 (k ) ). Denote the transmit power of the relay in frame k as p2 (k). Then, as the EA method is adopted, the achievable data rate in the cooperative transmission from the source and the relay to the destination can be expressed as

Rc (k ) = log (1 + P1 G3 (k ) + p2 (k )G2 (k ) ).

(1)

2.2. Channel model

Furthermore, for CT mode, we can obtain the data rate which can be achieved in the WPRN in frame k as [29]

We assume that the channels in the WPRN are quasi-static fading. That is, the channel coefficients remain constant in the du-

RCT (k ) = min{λ2 (k )R1 (k ), [λ1 (k ) + λ2 (k )]R3 (k ) + λ3 (k )Rc (k )}. (2)

G. Huang and W. Tu / Computer Networks 160 (2019) 65–76

Then, we can express the average throughput over one period, which includes K consecutive frame transmissions, for the WPRN as

T =

1 K

K 

[(1 − θ (k ))RDT (k ) + θ (k )RCT (k )].

(3)

k=1

2.4. Energy scheduling constraint at the relay With the HSC model, we can schedule the energy consumption across multiple frames at the relay when the WPRN operates in CT mode. Denote the energy utilization efficiency as η ∈ (0, 1). Then, when the WPRN operates in DT mode in frame k, the amount of harvested energy is calculated as EDT (k ) = ηT f P1 g1 (k ). When the WPRN operates in CT mode in frame k, the relay harvests energy in the first phase, and the amount of harvested energy is calculated as ECT (k ) = ηλ1 (k )T f P1 g1 (k ). Thus, in frame k, the amount of available energy can be calculated as [29]

E (k ) = (1 − θ (k ))EDT (k ) + θ (k )ECT (k ) = V (k )[(1 − θ (k ) ) + θ (k )λ1 (k )],

(4)

where V (k ) = ηT f P1 g1 (k ). Denote the time instant just before frame k as k− . Furthermore, at time instant k− , denote the amount of available energy at the rechargeable device as B(k) ≥ 0, which can be measured at the relay. Then, the energy scheduling constraint at the relay in frame k can be written as [29] k 



T f θ (t )λ3 (t ) p2 (t ) ≤ min B(1 ) +

t=1

k 



E (t ), Bmax ,

(5) where Bmax is the maximum allowable stored energy in the rechargeable device at the relay. Remark 1. As mentioned previously, we ignore the circuit power at the relay in this paper. When the circuit power is taken into account, the energy available for information relaying decreases, which causes the achievable end-to-end data rate of CT mode decreases, and the WPRN may opt to operate in DT mode in more frames within one time period. To obtain the corresponding transmission strategy, we just need to modify the energy scheduling constraint in (5) as

t=1

T f θ (t )[λ3 (t ) p2 (t ) + Pc ]



≤ min B(1 ) +

k 

In this paper, we aim at maximizing the average throughput, which is defined in (3), by optimizing the TS operation and power allocation over one period. We denote the variables to be optimized as (k ) = {θ (k ) ∈ {0, 1}, p2 (k ) ≥ 0, λ(k ) ∈ (k )}. As the full CSI of K frames in one period has been known, we can formulate the optimization problem as

max

(k ),∀k∈K

K 

[(1 − θ (k ))RDT (k ) + θ (k )RCT (k )]

(7a)

k=1

s.t. ( 5 ).

(7b)

The energy scheduling constraint (7b) in problem (7) is nonconvex due to the product term in the left side. Moreover, problem (7) is coupled over K frames. Thus, problem (7) is hard to solve directly. To make it tractable, we denote a vector of length k as X k = [X (1 ), · · · , X (k )] in general, e.g., the energy of the rechargeable device at the relay from frame 1 to frame k is expressed as Bk = [B(1 ), · · · , B(k )]. Then, at frame (k + 1 )− , the stored energy is updated in general as

B(k + 1 ) = f (Bk , pk2 , λk ).

(8)

Then, we can also rewrite the energy scheduling constraint (7b) as

T f θ (k )λ3 (k ) p2 (k ) ≤ min {B(k ) + E (k ) − B(k + 1 ), Bmax }

(9)

for k ∈ Ko and

(10)

We obtain (10) because the relay will use up all the available energy to transmit when the WPRN operates in CT mode. Note that B(k) (k ∈ K) can be measured at the relay before the transmission of frame k, while B(k + 1 ) (k ∈ Ko) is a variable to be determined since p2 (k) needs to be optimized. Moreover, according to (9), we have

B ( k ) + E ( k ) − B ( k + 1 ) ≥ 0,

(11)

otherwise the constraint in (9) cannot be satisfied. Thus, we obtain (a )

B ( k + 1 ) ≤ B ( k ) + E ( k ) ≤ B ( k ) + V ( k ),

(12)

where (a) holds because we can obtain E(k) ≤ V(k) according to (4).  Thus, we let  (k ) = (k ) B(k + 1 ), for k ∈ Ko. Then, according to Bellman equation [32], we can transform problem (7) into the following equations which can equivalently obtain the optimal solutions,

F ∗ (B(k ), k ) = max(1 − θ (k ))RDT (k ) + θ (k )RCT (k )

 E (t ), Bmax ,

3.1. Problem formulation

T f θ (K )λ3 (K ) p2 (K ) ≤ min {B(K ) + E (K ), Bmax }.

∀k ∈ K,

t=1

k 

69

∀k ∈ K,

 ( k )

+ F ∗ (B(k + 1 ), (k + 1 ) ),

(6)

t=1

where the constant Pc is the circuit power consumption at the relay when the WPRN operates in CT mode. Since only a constant is appended in the constraint, it can be easily verified that the derivation of the optimization algorithms presented in Sections 3 and 4 can be extended to the scenario where the circuit power is taken into account. 3. Offline joint TS operation and power allocation algorithm In this section, we assume that the full CSI of K frames over one period has been known before transmission. Although it is an ideal scenario, the derived offline joint TS operation and power allocation algorithm can provide a benchmark for the practical scenario where only causal CSI is known.

s.t. ( 9 ),

∀k ∈ K o , (13)

F ∗ (B(K ), K ) = max(1 − θ (K ))RDT (K ) + θ (K )RCT (K ) s.t. (10 ). (K )

(14) To obtain the optimal value of problem (7) by the above equations, we need to solve these equations recursively by using the backward induction method [32]. That is, start from the last Eq. (14) and work backward until the first equation in (13) for k = 1. Then, the obtained F∗ (B(1), 1) is the same as the optimal value in problem (7). To solve the equations in (13) and (14), we first derive the decision criteria of optimal network operation mode, i.e., θ (k) (k ∈ K), and then obtain the solutions of the other optimized variables in  (k) (k ∈ Ko) and (K).

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3.2. Decision criteria of network operation mode Recall that the relay performs EH when the WPRN operates in DT mode. Thus, given the energy state B(k) at time instant k− , the energy state at time instant (k + 1 )− can be calculated as B(k + 1 ) = B(k ) + V (k ). Note that B(k + 1 ) ∈ B. Thus, B(k + 1 ) should be obtained as the optimal solution, i.e., x∗ , to the problem such as follows,

min B(k ) + V (k ) − x, s.t. B(k ) + V(k ) − x ≥ 0. x∈B

(15)

¯ k ) = (k )\θ (k ) We denote B˜(k ) = x∗ . Furthermore, define ( ¯  (k ) =  (k )\θ (k ). Then, the WPRN can determine its optiand  mal operation mode as stated in the following proposition. Proposition 1. If g3 (k) ≥ g1 (k), the optimal operation mode of the WPRN in frame k (k ∈ K) is θ ∗ (k ) = 0. Otherwise, θ ∗ (k) (k ∈ K) is determined by



θ ∗ (k ) =

0, ( B ( k ), k ) ≥ 1, otherwise FD∗

FC∗

( B ( k ), k )

in FD∗ (B(K ), K ) = RDT (K ),

which  F ∗ B˜(k + 1 ), (k + 1 ) ,

(16) FD∗ (B(k ), k ) = RDT (k ) +

FC∗ (B(K ), K ) = max RCT (K )

(17a)

s.t. T f λ3 (K ) p2 (K ) ≤ B(K ) + λ1 (K )V (K )

(17b)

¯ K) (

by solving problem (20) with the brute force apB∗ (k + 1 ) in B proach. However, because the energy scheduling constraint (18b) is non-convex, problem (19) is non-convex. Thus, it is still not easy to solve problem (19) directly. To tackle this problem, we propose to solve it by a two-level optimization approach, namely an inner-lever problem and an outer-level problem. The inner-level

∗ (k ), where problem is to fix λ1 (k) and search for the optimal 

¯ (k ) = (k )\λ1 (k ), to the following optimization problem

A∗ (λ1 (k ) ) = max RCT (k ) s.t. (18b ).

(21)

k) (

The outer-level problem is to search for the optimal λ∗1 (k ) to the following optimization problem

max

λ1 (k )∈[0,1]

A∗ (λ1 (k )).

(22)

Note that the right-hand side of (18b) (denoted as W (k ) = B(k ) + λ1 (k )V (k ) − B(k + 1 )) may be negative for some given B(k + 1 ) ∈ B  and λ1 (k) ∈ [0, 1]. In this case, problem (21) has no feasible solution and we let the optimal value of (21) as A∗ (λ1 (k ) ) = 0. Otherwise, we have the following lemma for the achievable data rates in the two phases of one frame transmission in CT mode. Lemma 1. Given λ1 (k) ∈ [0, 1], the following inequality

λ2 (k )R1 (k ) ≥ [λ1 (k ) + λ2 (k )]R3 (k ) + λ3 (k )Rc (k )

and

(23)

is satisfied when the solution to problem (21) is achieved at optimality.

FC∗ (B(k ), k ) = max RCT (k ) + F ∗ (B(k + 1 ), (k + 1 ) ), ∀k ∈ Ko, (18a)

Proof. See Appendix B.

s.t. T f λ3 (k ) p2 (k ) ≤ B(k ) + λ1 (k )V (k ) − B(k + 1 ).

According to Lemma 1, we can establish the following proposition.

¯  (k ) 

(18b) 

Proof. See Appendix A.

To determine the network operation mode by Proposition 1, we need to solve problem (17) and problem (18). Note that if we solve problem (18), we can also solve problem (17) by using the same approach. Thus, we only focus on solving problem (18) in the following subsection, which corresponds to the resource allocations in CT mode. 3.3. Resource allocations in CT mode

Q ∗ (B(k + 1 )) = max RCT (k ) s.t. (18b ), ¯ k) (

Proposition 2. Given λ1 (k) ∈ [0, 1] and suppose that W(k) ≥ 0. When the solution to problem (21) is achieved at optimality, the energy scheduling constraint (18b) is active. That is,

T f λ3 (k ) p2 (k ) = W (k ).

(24)

Proof. See Appendix C.



According to (24) in Proposition 2, we can obtain the optimal p2 (k) as



Problem (18) is still not easy to solve because the variable B(k + 1 ) is continuous, which makes the number of possible solutions is infinite, and the problem is coupled with two frames, i.e., frame k and frame (k + 1 ). To make it tractable, as in [30], we adopt a discrete model for the energy states of the rechargeable device. That is, the number of energy states is finite, which is denoted as (L + 1 ), and the set  of energy states can be de2Bmax fined as B  0, Bmax , , · · · , B . Then, B(k + 1 ) should satmax L L isfy B(k + 1 ) ∈ B and the inequality (12). Thus, we can obtain a . feasible set of B(k + 1 ), which is defined as B Based on the discrete energy state model, we can decompose problem (18) into two subproblems, namely an inner problem and an outer problem. The inner problem is obtained by fixing B(k + in problem (18), i.e., 1) ∈ B

(19)

while the outer problem is to obtain the optimal B∗ (k + 1 ), i.e., ∗ FCT (B(k ), k ) = max Q ∗ (B(k + 1 )) + F ∗ (B(k + 1 ), (k + 1 ) ). (20) B(k+1 )∈B

We can obtain the optimal value of problem (18) by two steps. The first step is to obtain Q ∗ (B(k + 1 )) by solving problem (19) for all B(k + 1 )’s, and the second step is to search for the optimal



p2 ( k ) =

W (k ) T f λ3 (k )

+

.

(25)

Then, we can rewrite problem (21) as

A∗ (λ1 (k )) =

max RCT (k ),

(26)

λ3 (k )∈[0,1]





where RCT (k ) = min λ2 (k )R1 (k ), [λ1 (k ) + λ2 (k )]R3 (k ) + λ3 (k )Rc (k ) ,



in which Rc (k ) = λ3 (k ) log 1 + P1 G3 (k ) +



W (k ) T f λ3 ( k )

+



G2 ( k ) .

It can be proved that RCT (k ) is concave in λ3 (k). Therefore, problem (26) can be solved by the means of the golden section search technique. Given λ1 (k) and according to λ2 (k ) = 1 − λ1 (k ) − λ3 (k ), the involved algorithm to solve problem (26) is illustrated in Algorithm 1. Note that A∗ (λ1 (k)) is concave. Therefore, after achieving the optimal value of (26), we can also employ the golden section search technique to solve problem (22). As a result, we illustrate the involved algorithm to solve the inner problem (19) in Algorithm 2. In summary, as the knowledge of full CSI is available, we can obtain the joint TS operation and power allocation algorithm to achieve the maximum throughput in the WPRN as Algorithm 3, where the input is B(1) and gi (k), ∀i ∈ {1, 2, 3}, ∀k ∈ K.

G. Huang and W. Tu / Computer Networks 160 (2019) 65–76

Algorithm 1 Algorithm to Solve Problem (26). 1: 2: 3: 4: 5: 6: 7: 8: 9:

Initialization: Set tolerance  > 0, λmin = 0, λmax = 1, λU = 3 3 3 L = λmin . λmax and λ 3 3 3 while λU − λL3 >  do 3 λU3 = λmin + (λmax − λmin 3 3 ) ∗ 0.618, 3 L max min λ3 = λ3 + λ3 − λU3 , Use λ3 (k ) = λU and λ3 (k ) = λL3 to calculate RCT (k ) to obtain 3 RCT,U (k ) and RCT,L (k ), respectively, if RCT,U (k ) > RCT,L (k ) then

λmin = λL3 , 3

else

λmax = λU . 3 3

end if end while + λL3 )/2 andλ∗2 (k ) = 1 − λ1 (k ) − λ∗3 (k ), 12: Calculate λ∗3 (k ) = (λU 3 ∗ calculate p2 (k ) according to (25) and obtain A∗ (λ1 (k )). 10: 11:

Algorithm 2 Algorithm to Solve the Inner Problem (19). 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22:

Initialization: Set tolerance ε > 0, λmin = 0, λmax = 1, λU = 1 1 1 L = λmin . λmax and λ 1 1 1 while λU − λL1 > ε do 1 λU1 = λmin + (λmax − λmin ) ∗ 0.618, 1 1 1 min − λU , λL1 = λmax + λ 1 1 1 Use λ1 (k ) = λU to calculate W (k ) (denoted as WU (k )), and 1 use λ1 (k ) = λL1 to calculate W (k ) (denoted as WL (k )). if WU (k ) < 0 then A∗U (λ1 (k )) = 0. else Employ Algorithm 1 to obtain A∗ (λ1 (k )) (denoted as A∗U (λ1 (k ))) with λ1 (k ) = λU . 1 end if if WL (k ) < 0 then A∗L (λ1 (k )) = 0. else Employ Algorithm 1 to obtain A∗ (λ1 (k )) (denoted as A∗L (λ1 (k ))) with λ1 (k ) = λL1 . end if if A∗U (λ1 (k )) > A∗L (λ1 (k )) then λmin = λL1 . 1 else λmax = λU . 1 1 end if end while Obtain λ∗1 (k ) = (λU + λL1 )/2. 1

As mentioned previously, we cannot achieve the performance by Algorithm 3 in practice, because it requires full CSI, which is non-causal, over one time period before transmission. Nevertheless, we provide a theoretical upper bound on performance of the proposed practical strategy which requires only causal CSI, which will be derived in the next section. 4. Online joint TS operation and power allocation algorithm In the previous section, we have derived the offline joint TS operation and power allocation algorithm by assuming that the knowledge of full CSI is available, which can achieve an upper bound of the throughput performance in the WPRN. However, in practice, a transmitter cannot know the CSI in subsequent frames before it transmits in some frame. That is, the transmitter has only the causal knowledge of the CSI. Therefore, in this section, we investigate the online joint TS operation and power allocation

71

Algorithm 3 Algorithm to Achieve the Maximum Throughput in the WPRN with Full CSI. 1: Set l = 0. 2: while l ≤ L do Calculate B(K ) = l Bmax and FD∗ (B(K ), K ) = RDT (K ). 3: L if g3 (K ) ≥ g1 (K ) then 4: Set θ ∗ (K ) = 0. 5: else 6: ¯ ∗ (K ) by solving problem (17), Obtain FC∗ (B(K ), K ) and  7: ∗ Obtain θ (K ) according to (16). 8: 9: end if if θ ∗ (K ) = 0 then 10: Let F ∗ (B(K ), K ) = FD∗ (B(K ), K ) and ∗ (K ) = {θ ∗ (K )}. 11: else 12: Let F ∗ (B(K ), K ) = FC∗ (B(K ), K ) and derive ∗ (K ) from 13: ¯ ∗ ( K ).  end if 14: Record ∗ (K )|B(K ) = ∗ (K ) and set l = l + 1. 15: 16: end while 17: Set k = K and l = 0. 18: while k ≥ 1 do k = k − 1. 19: 20: while l ≤ L do Calculate B(k ) = l Bmax and FD∗ (B(k ), k ) = RDT (k ). 21: L if g3 (k ) ≥ g1 (k ) then 22: Set θ ∗ (k ) = 0. 23: else 24: ¯ ∗ (k ) by solving problem (18), Obtain FC∗ (B(k ), k ) and  25: ∗ Obtain θ (k ) according to (16). 26: 27: end if if θ ∗ (k ) = 0 then 28: 29: Let F ∗ (B(k ), k ) = FD∗ (B(k ), k ) and ∗ (k ) = {θ ∗ (k )}. else 30: Let F ∗ (B(k ), k ) = FC∗ (B(k ), k ) and derive ∗ (k ) from 31: ¯ ∗ ( k ).  end if 32: Record ∗ (k )|B(k ) = ∗ (k ) and set l = l + 1. 33: 34: end while 35: end while 36: Deduce the throughput based on ∗ (1 )|B(1 ) .

algorithm when the CSI is known causally. Moreover, to utilize the wireless channel variations to improve the throughput performance, we also assume that the pdfs of the channels in the WPRN have been known at the transmitters. To obtain the algorithm, we first formulate the involved problem as a stochastic optimization problem. Then, we adopt a FSMC model [31] to transform the formulated problem into a deterministic form, where an online algorithm with low computational complexity is derived to achieve superior throughput performance in the WPRN. 4.1. Problem formulation To formulate the throughput maximization problem when only causal CSI is available, we denote the CSI in the WPRN in frame k as g(k ) = {g1 (k ), g2 (k ), g3 (k )}. Moreover, considering the energy state, i.e., B(k), we define the state of the WPRN in frame k as D(k ) = {g(k ), B(k )}. Then, because the CSI in subsequent frames is random and cannot be known in advance, the average throughput over a period expressed in (3) should be modified as





K 1 T¯ = E [(1 − θ (k ))RDT (k ) + θ (k )RCT (k )]|D(1 ) , K k=1

(27)

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G. Huang and W. Tu / Computer Networks 160 (2019) 65–76

where the expectation E{·} is performed over all g(k)’s in a period. Thus, we can modify the throughput maximization problem in (7) as



max E

(k ),∀k∈K

K 



[(1 − θ (k ))RDT (k ) + θ (k )RCT (k )]|D(1 )

(28a)

k=1

s.t. ( 5 ).

(28b)

Recall that the energy scheduling constraint (5) can be rewritten as (9) and (10). Thus, similar to that in Section 3.1, we can transform problem (28) into the following equations

F ∗ (D(k ), k ) = max(1 − θ (k ))RDT (k ) + θ (k )RCT (k )  ( k )

+ F¯ ∗ (D(k + 1 ), (k + 1 ) ),

∀k ∈ K o ,

s.t. ( 9 ),

(29)

F ∗ (D(K ), K ) = max(1 − θ (K ))RDT (K ) + θ (K )RCT (K ) s.t. (10 ), (K )

(30) where

F¯ ∗ (D(k + 1 ), (k + 1 ) ) = Eg(k+1) [F ∗ (D(k + 1 ), (k + 1 ) )].

(31)

4.2. Problem transformation with the FSMC model Although the statistics of g(k + 1 ) is known, it is not easy to obtain F¯ ∗ (D(k + 1 ), (k + 1 ) ) in (34). Therefore, the equations in (29) and (30) are more difficult to solve than those in (13) and (14). Thus, in the follows, we adopt a FSMC model to transform these equations into deterministic forms, which can make the involved problems tractable. The FSMC models have been widely used for modeling wireless flat-fading channels in a variety of applications since the mid 1990s, which are versatile, and with suitable choices of model parameters, can capture the essence of time-varying fading channels [31]. By employing the FSMC models, the continuous channel gain can be quantized with a finite number of values, where each value is named as a state. There have been many methods to quantize the channel gains into a finite number of states [31]. Specially, we adopt the equal probable steady state method [31] in this paper, where the steady-state probabilities for all obtained FSMC states are equal. Suppose that the channel gain g(k) is quantized into N states, which are included in a discrete set χ (k ) = {g1 (k ), g2 (k ), · · · , gN (k )}, and denote the state of g(k) as ξ (k ) = {ξ1 (k ), ξ2 (k ), ξ3 (k )}, where ξ (k) ∈ χ (k). Furthermore, let D˜ (k ) = {ξ (k ), B(k )}. Then, we can rewrite the equations in (29) and (30) as







4.3. Joint time switching and power allocation algorithm with causal CSI As the equations in (32) and (33) can be solved, we can obtain the joint TS operation and power allocation strategy for a given ˜ (k ), ∀k ∈ K. Note that for D ˜ (k ), the number of possible state D ˜ ( 1 ), · · · , D ˜ ( K )} values can be N 3 × (L + 1 ). Furthermore, for S = {D over one period, the number of possible values can be K × N 3 × (L + 1 ). Therefore, we can establish a look-up table which has K × N 3 × (L + 1 ) entries, where an entry corresponds to a state ˜ (k ). In the entry corresponding to D ˜ (k ) (k ∈ Ko), we record the D ˜ (K ), we optimal {θ ∗ (k ), B∗ (k + 1 )}. In the entry corresponding to D record {θ ∗ (K)}. We illustrate the procedure to obtain the optimal network operation mode θ ∗ (k) and energy state B∗ (k + 1 ) for each ˜ (k ) in Algorithm 4. Note that when θ ∗ (k ) = 0 and k ∈ Ko, state D we set B∗ (k + 1 ) = NULL in the record because the relay only harvests energy and does not need to know the energy state B∗ (k + 1 ) when the WPRN operates in DT mode. Algorithm 4 The Procedure to Obtain the Optimal Network Operation Mode and Energy State under the FSMC Model. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19:

˜ (k ), k = max(1 − θ (k ))R˜DT (k ) + θ (k )R˜CT (k ) F∗ D

20:

∀k ∈ K o ,   s.t. Tf θ (k )λ3 (k )p2 (k ) ≤ min B(k ) + E˜ (k ) −B(k + 1 ), Bmax , (32)

22:

 ( k )



21:



˜ ( k + 1 ), ( k + 1 ) , + F¯ ∗ D

F

∗



(K )





s.t. Tf θ (K )λ3 (K )p2 (K ) ≤ min B(K ) + E˜ (K ), Bmax ,

(33)

where R˜DT (k ), R˜CT (k ) and E˜ (k ) can be obtained by replacing g(k) with ξ (k) in RDT (k), RCT (k) and E(k), respectively. As the equal probable steady state method is employed,



 ˜ ( k + 1 ), ( k + 1 ) = we obtain p ξ (k + 1 ) = N13 . Note that F¯ ∗ D 

 ∗

  ˜ ξ (k+1 ) p ξ (k + 1 ) F D(k + 1 ), (k + 1 ) . Thus, we have



23: 24: 25:

˜ (K ), K = max(1 − θ (K ))R˜DT (K ) + θ (K )R˜CT (K ) D



˜ ( k + 1 ), ( k + 1 ) = F¯ ∗ D

1  N3

ξ (k+1 )

 ∗

F



˜ ( k + 1 ), ( k + 1 ) . D

(34)



˜ (k + 1 ), (k + 1 ) has been derived, we can solve the As F¯ ∗ D equations in (32) and (33) by the same approach that we have derived to solve the equations in (13) and (14).

Set l = 0. while l ≤ L do Calculate B(K ) = l Bmax and FD∗ (B(K ), K ) = RDT (K ). L for all possible channel state ξ (K ) do Solve equation (33) according to step 4–14 in Algorithm 3. ˜ ( K ). Record {θ ∗ (K )} in the entry corresponding to D end for

 ˜ (K ), K according to (34). Calculate F¯ ∗ D Set l = l + 1. end while Set k = K and l = 0. while k ≥ 1 do k = k − 1. while l ≤ L do Calculate B(k ) = l Bmax and FD∗ (B(k ), k ) = RDT (k ). L for all possible channel state ξ (k ), do Solve equation (32) according to step 22–32 in Algorithm 3. Record {θ ∗ (k ), B∗ (k + 1 )} in the entry corresponding to ˜ ( k ). D end for if k > 1 then

 ˜ (k ), k according to (34). Calculate F¯ ∗ D end if Set l = l + 1. end while end while

Note that we can establish the look-up table offline before transmission. After the establishment of the look-up table, we can implement joint TS operation and power allocation online for the WPRN. We present the involved procedure in Algorithm 5. In this algorithm, the input includes the look-up table and the network state D(k ) = {g(k ), B(k )}. Therefore, Algorithm 5 is based on causal CSI, and thus it is a practical algorithm. In step 6 of Algorithm 5, we solve problem (19) to obtain the joint TS operation and power allocation strategy. This is because the inner problem of problem (29) is the same as (19).

G. Huang and W. Tu / Computer Networks 160 (2019) 65–76

Algorithm 5 Algorithm. 1: 2: 3: 4: 5: 6: 7: 8: 9:

Online Joint Time Switching and Power Allocation

˜ ( k ). Map the CSI g(k ) to ξ (k ) to obtain D ˜ (k ), and read θ ∗ (k ) Search for the entry corresponding to D from the located entry. if θ ∗ (k ) = 0 then The source sends information directly to the destination and the relay harvests energy from the RF signals. else Read B∗ (k + 1 ) from the located entry, Let B(k + 1 ) = B∗ (k + 1 ) and solve problem (19) to achieve ¯ ∗ ( k ), the optimal  The source and the relay cooperatively send information to ¯ ∗ ( k ). the destination based on the optimal  end if

4.4. Complexity analysis The computational complexity of Algorithm 5 is mainly from solving problem (19), which is implemented by Algorithm 2. In Algorithm 2, the computational complexity is from implementing the two-layer golden section search technique. For the golden section search technique, we denote the computational complexity as X. Thus, the computational complexity to implement two-layer golden section search is O{X 2 }, which indicates that Algorithm 5 has low computational complexity. Note that the offline algorithm, i.e., Algorithm 3, has higher computational complexity than the online algorithm, i.e., Algorithm 5. This is because the computational complexity of Algorithm 3 is mainly caused by solving problem (17) for (L + 1 ) discrete energy states and problem (18) for (L + 1 ) discrete energy states over (K − 1 ) consecutive frames. To solve problem (17) or (18), Algorithm 2 is required to be executed. Thus, the computational complexity of Algorithm 3 is O{(L + 1 )KX 2 }. Nevertheless, Algorithm 3 can be implemented offline by a general-purpose computer, which has higher computational capacity than the wireless nodes in the WPRN, thus the higher computational complexity can be easily tackled in practice. 5. Simulation results We evaluate the performance of our proposed scheme by computer simulations in this section. We assume that the time-varying i.i.d. AWGN channel has zero mean and unit variance. The source-

73

to-destination distance is d3 = 10m, the source-to-relay distance is d1 = ρ d3 (0 < ρ < 1) and the relay-to-destination distance is d2 = d3 − d1 . We assume that the channel between the source and the relay and the channel between the relay and the destination follow the line-of-sight propagation, thus we set the power pathloss exponents as η = 2. Meanwhile, we assume that there are obstacles between the source and the destination, thus we set the power path-loss exponent as η = 4. As in [26], we adopt a pathloss model which is expressed as η (10 + 10 log10 di ), where i ∈ {1, 2, 3}. We set the power of AWGN as σ22 = σ32 = 2 × 10−9 W, the energy utilization efficiency as τ = 0.8, B(1 ) = 0, the number of frames in one period as K = 10 and the duration of one frame as T f = 10μs. Moreover, we set the number of energy states as L = 20 and let Bmax = 2Eh = 2τ P1 g¯ 1 , where g¯ 1 is the mean of g1 . In the follows, we refer to the proposed scheme as the practical scheme which is implemented by Algorithm 5. To show the superior performance achieved by our proposed scheme, we compare the throughput achieved by our proposed scheme with that achieved by the existing schemes in Figs. 5 and 6. In Fig. 5, we set P1 = 20dBm and show the achieved throughput when the relay location varies. Meanwhile, in Fig. 6, we set ρ = 0.5 and show the achieved throughput when the transmit power of the source, i.e., P1 , varies. From both the two figures, it can be seen that our proposed scheme outperforms the conventional TSR-based HTF scheme, the direction transmission scheme and the ATF scheme. Moreover, it can be seen that the gap between the throughput achieved by the proposed practical scheme and its upper bound, which is achieved by Algorithm 3, is very small. Furthermore, to reveal the advantages of enabling the relay to harvest energy in CT mode, we illustrate the throughput achieved by a simple scheme without EH in CT mode, where we fix λ1 (k ) = 0 while solving problem (19), which indicates that the relay harvests energy only when the WPRN operates in DT mode. It can be seen from Figs. 5 and 6 that our proposed scheme outperforms the proposed simple scheme, which indicates that the throughput performance can be further improved when the relay is enabled to harvest energy in CT mode. In Fig. 7, we show the impact of the number of frames, i.e., K, in one period on the throughput performance in our proposed scheme., where ρ = 0.5 and P1 = 20dBm. From Fig. 7, it can be found that the achieved throughput can be improved as K increases. The reason is that more wireless channel variations can be utilized as K becomes larger. Specially, it can be found in Fig. 7 that the achieved throughput increases distinctly when K is small and becomes steady when K is large. Note that larger K results in more entries in the look-up table, which indicates that higher

Fig. 5. The average throughput achieved by different schemes when the relay locates differently.

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G. Huang and W. Tu / Computer Networks 160 (2019) 65–76

Fig. 6. The average throughput achieved by different schemes when the transmit power of the source varies.

Fig. 7. Average throughput achieved by our proposed scheme with different K values.

Fig. 8. The impact of the direct link on the throughput performance.

computational complexity is required for the establishment of the look-up table. Therefore, K should be set as a medium value when implementing the proposed scheme in practice, which is the reason why we set K = 10 in the other simulations in this paper. Fig. 8 shows the achieved throughput by varying the path loss exponent of the direct link when implementing Algorithm 5. To verify the impact of the direct link on throughput performance,

we also illustrate the achieved throughput where the direct link is ignored and the throughput achieved by direct transmission in Fig. 8. From Fig. 8, it can be found that the throughput can be significantly improved by considering the direct link, especially when the path loss exponent, i.e., η, of the direct link varies from 2 to 4. Meanwhile, it can be found that the deployment of EH relay can increase the throughput since the throughput achieved by the

G. Huang and W. Tu / Computer Networks 160 (2019) 65–76

75

Fig. 9. Average throughput achieved by our proposed scheme with different Bmax values and L values.

proposed scheme with direct link is higher than that achieved by direct transmission, especially when the path loss exponent of the direct link varies from 3 to 5. According to the analysis in previous sections, we have quantized the energy state of the rechargeable device into discrete values with the step length Bmax in our proposed scheme. Clearly, the L smaller the step length and the larger Bmax are, the more accurate (or higher) the throughput can be achieved. Therefore, to evaluate the impact of the values of Bmax and L on the performance in our proposed scheme, we show the throughput by setting Bmax and L with different values in Fig. 9. From Fig. 9, it can be found that when Bmax = 2Eh , the throughput achieved by setting L = 20 is almost the same as that achieved by setting L = 100. Moreover, it can be found in Fig. 9 that although the throughput can be improved by increasing Bmax from 2Eh to 10Eh and setting L = 100, the achieved throughput gains are small. Note that as the number of energy states increases, the computational complexity of the establishment of the look-up table increases. Therefore, we set Bmax = 2Eh and L = 20 in the previous simulations.

design of optimization scheme for the multi-relay WPRN is an interesting future work Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grant 61872098, the Natural Science Foundation of Guangdong Province under Grants 2017A030313363 and 2017A030310639, the Science & Technology Project of Guangdong Province under Grant 2016A010101032, the Science & Technology Project of Guangzhou City under Grant 201605030014 and the Academic Visiting Scholarship under the State Scholarship Fund of China. Appendix A. Proof of Proposition 1

6. Conclusions In this paper, we investigated the WPRN with a direct link in order to improve the throughput performance by the joint optimization of time switching and power allocation for multiple frames. By considering the DT and CT network operation modes and the HSC energy consumption model, we formulated the problems to maximize throughput for both the cases of full CSI and causal CSI. First, we used dynamic programming and problem decomposition techniques to derive an algorithm when solving the problem for the case of full CSI. This provided a benchmark of achievable throughput under the HSC model. Motivated by the derivation of the results for the case of full CSI and by using the FSMC channel model, we then solved the problem for the case of causal CSI and proposed an online and low-complexity algorithm to achieve joint time switching and power allocation. Our simulation results verified our scheme achieved the better performance in terms of throughput over a period of time which was for transmissions of consecutive multiple frames. Note that although this paper focuses on a WPRN with a single reply (as those in [10–24]), the results may provide insights into the design of more complex WPRNs. When multiple relays exist, we can further enhance the network performance by jointly optimizing the power allocations at the relays while the relays operate in CT mode, which is a more complicated problem. Therefore, the

Proposition 1 consists of two parts. The first part is for the case of g3 (k) ≥ g1 (k), and the second part is for the case of g3 (k) < g1 (k). Since FD∗ (B(k ), k ) and FC∗ (B(k ), k ) can be obtained by substituting θ (k ) = 0 and θ (k ) = 1 into Eqs. (13) and (14), respectively, the second part can be easily verified by comparing the values of FD∗ (B(k ), k ) and FC∗ (B(k ), k ), where k ∈ K. Therefore, we focus on the proof of the first part in the follows. According to (14), we have F ∗ (B(K ), K ) = FD∗ (B(K ), K ) when θ (K ) = 0 and F ∗ (B(K ), K ) = FC∗ (B(K ), K ) when θ (K ) = 1. Meanwhile, according to (13), we have F ∗ (B(k ), k ) = FD∗ (B(k ), k ) when θ (k ) = 0 and F ∗ (B(k ), k ) = FC∗ (B(k ), k ) when θ (k ) = 1, where k ∈ Ko . When g3 (k) ≥ g1 (k), according to the expression of RCT (k) in (2), ¯ K ) and RDT (k) ≥ RCT (k) for any we obtain RDT (K) ≥ RCT (K) for any ( ¯  (k ). Since RDT (K) ≥ RCT (K) for any ( ¯ K ), we have F ∗ (B(K ), K ) >  D ∗ ∗ FC (B(K ), K ). Thus, the optimal θ (K) to problem (30) is θ ∗ (K ) = 0. Let’s consider the case of k ∈ Ko. According to (15), we obtain B˜(k + 1 ) ≥ B(k + 1 ) when k ∈ Ko. Meanwhile, the F∗ (·) function in Eqs. (13) and (14) increases monotonically with B(k). Therefore,  we can obtain F ∗ B˜(k + 1 ), (k + 1 ) ≥ F ∗ (B(k + 1 ), (k + 1 ) ) when k ∈ Ko. Moreover, when g3 (k) ≥ g1 (k), since RDT (k) ≥ RCT (k) for any ¯  (k ), we can obtain that F ∗ (B(k ), k ) > F ∗ (B(k ), k ), which results  D C in that θ ∗ (k ) = 0. To this end, the proof of Proposition 1 is completed.

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Appendix B. Proof of Lemma 1 We prove this lemma by reductio ad absurdum. Suppose that at optimality we have

λ2 (k )R1 (k ) < [λ1 (k ) + λ2 (k )]R3 (k ) + λ3 (k )Rc (k ).

(35)

In this case, we can decrease λ3 (k) such that the inequality in (35) and the constraint in problem (21), i.e., (18b) are still satisfied. As we decrease λ3 (k), we can increase the value of λ1 (k ) + λ2 (k ). Since λ1 (k) is given, the left hand side of (35), i.e., λ2 (k)R1 (k), increases. As a result, the objective function in problem (21) increases. This contradicts our initial assumption that the achieved solution is at optimality. Appendix C. Proof of Proposition 2 We prove this proposition by reductio ad absurdum. Suppose that at optimality we have Tf λ3 (k)p2 (k) < W(k). Then, we can increase p2 (k) such that T f λ3 (k ) p2 (k ) = W (k ). As a result, the lefthand side (23), i.e., λ2 (k)R1 (k), does not decrease, and the righthand side, i.e., [λ1 (k ) + λ2 (k )]R3 (k ) + λ3 (k )Rc (k ), increases. From Lemma 1, we obtain that at optimality the right-hand side of (23) determines the optimal value of the objective function in (21). Therefore, the optimal value of problem (21) increases. This contradicts our initial assumption that the achieved solution is at optimality. References [1] Y. Xu, C. Shen, Z. Ding, X. Sun, S. Yan, G. Zhu, Z. Zhong, Joint beamforming and power-splitting control in downlink cooperative SWIPT NOMA systems, IEEE Trans. Signal Process. 65 (18) (2017) 4874–4886. [2] C. Xu, C. Song, P. Zeng, H. Yu, Secure resource allocation for energy harvesting cognitive radio sensor networks without and with cooperative jamming, Comput. Netw. 141 (2018) 189–198. [3] I. Ahmed, M.M. Butt, C. Psomas, A. Mohamed, I. Krikidis, M. Guizani, Survey on energy harvesting wireless communications: challenges and opportunities for radio resource allocation, Comput. Netw. 88 (2015) 234–248. [4] G. Zhang, J. Xu, Q. Wu, M. Cui, X. Li, F. Lin, Wireless powered cooperative jamming for secure OFDM system, IEEE Trans. Veh. Technol. 67 (2) (2018) 1331–1346. [5] Q. Li, Q. Zhang, J. Qin, Secure relay beamforming for SWIPT in amplify-and– forward two-way relay networks, IEEE Trans. Veh. Technol. 65 (11) (2016) 9006–9019. [6] D. Mishr, G.C. Alexandropoulos, S. De, Transmit precoding and receive power splitting for harvested power maximization in MIMO SWIPT systems, IEEE Trans. Green Commun. Netw. 2 (3) (2018) 774–786. [7] D. Mishra, G.C. Alexandropoulos, S. De, Energy sustainable IoT with individual QoS constraints through MISO SWIPT multicasting, IEEE Internet Things J. 5 (4) (2018) 2856–2867. [8] D. Mishra, G.C. Alexandropoulos, S. De, Jointly optimal spatial channel assignment and power allocation for MIMO SWIPT systems, IEEE Wireless Commun. Lett. 7 (2) (2018) 214–217. [9] D. Mishra, S. De, G.C. Alexandropoulos, D. Krishnaswamy, Energy-aware Mode Selection for Throughput Maximization in RF-Powered D2D Communications, in: IEEE Global Communications Conference (GLOBECOM), Singapore, 2017. [10] A.A. Nasir, X. Zhou, S. Durrani, R.A. Kennedy, Relaying protocols for wireless EH and information processing, IEEE Trans. Wireless Commun. 12 (7) (2013) 3622–3636. [11] Y. Feng, V.C.M. Leung, F. Ji, Performance study for SWIPT cooperative communication systems in shadowed nakagami-m fading channels, IEEE Trans. Wireless Commun. 17 (7) (2018) 1199–1211. [12] G. Amarasuriya, E.G. Larsson, H.V. Poor, Wireless information and power transfer in multiway massive MIMO relay networks, IEEE Trans. Wireless Commun. 15 (6) (2016) 3837–3855. [13] M. Mohammadi, B.K. Chalise, H.A. Suraweera, C. Zhong, G. Zheng, I. Krikidis, Throughput analysis and optimization of wireless-powered multiple antenna full-duplex relay systems, IEEE Trans. Commun. 64 (4) (2016) 1769–1785. [14] Y. Zhang, J. Ge, E. Serpedin, Performance analysis of a 5g energy-constrained downlink relaying network with non-orthogonal multiple access, IEEE Trans. Wireless Commun. 16 (12) (2017) 8333–8346.

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Wanqing Tu is a senior lecturer in the Department of Computing Science at The University of Auckland, New Zealand. She received her PhD from the Department of Computer Science at the City University of Hong Kong. Her research interests include wireless multi-hop communications, multimedia communications, QoS/QoE, IoT, overlay networks, and distributed computing. She received an IRCSET Embark Initiative Postdoctoral Research Fellowship. She received one Best Paper Award in 2005. She is a Senior Member of the IEEE.